Read the following scenario, which is represented by a polynomial expression. Then answer the questions to interpret the parts of the expression in terms of the given context.

Question

The volunteers at a high school football team’s concession stand are trying to decide on the price of the hot dogs they are selling. When they charge $2 for a hot dog, they sell an average of 70 hot dogs per game. With every $1 increase in the price, the number of hot dogs sold per game decreases by 8.

The volunteers can calculate the revenue earned from selling the hot dogs at each game using the expression -8x2 + 54x + 140, where x is the number of $1 increases in price.

Part A
What is the constant term in the polynomial expression, and what does it represent?

Answer 1

The constant term in the polynomial expression \(-8x^2 + 54x + 140\) is \(140\).

In the context of the scenario, this constant term represents the revenue earned from selling hot dogs when no price increase is applied (i.e., when \(x = 0\)). At that point, the price of the hot dogs is $2, and the average number sold is 70 hot dogs per game. Therefore, the revenue at this initial price is calculated as:

\[ \text{Revenue} = \text{Price} \times \text{Quantity Sold} = 2 \times 70 = 140. \]

So, the constant term \(140\) reflects the initial revenue before any price increase is implemented.